A274542 Decimal expansion of exp(sqrt(2)/3).

1, 6, 0, 2, 2, 4, 2, 9, 9, 7, 2, 0, 3, 5, 6, 0, 1, 5, 0, 9, 9, 5, 1, 7, 7, 7, 7, 2, 2, 2, 8, 6, 7, 8, 7, 5, 8, 5, 1, 2, 9, 6, 1, 6, 8, 2, 9, 5, 4, 5, 4, 7, 1, 8, 7, 4, 2, 6, 8, 2, 2, 4, 0, 5, 3, 0, 9, 1, 0, 0, 1, 6, 9, 9, 4, 9, 0, 4, 1, 9, 1, 9, 5, 8, 2

Offset: 1

Author

Johannes W. Meijer, Jun 27 2016

Keywords

• cons
• nonn

Comments

Define P(n) = (1/n)*(sum(x(n-k)*P(k), k=0..n-1)), n >= 1 and P(0) =1 with x(3) = (1 + sqrt(2)) and x(n) = 1 for all other n. We find that C2 = limit(P(n), n -> infinity) = exp(sqrt(2)/3).

The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.

Examples

			c = 1.6022429972035601509951777722286787585129616829545471874……
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• The Dev Team and Simon Plouffe, The Inverse Symbolic Calculator (ISC).
• Index entries for transcendental numbers

Crossrefs

Cf. A274540, A274541, A014176, A131594, A036039.

Programs

Magma

SetDefaultRealField(RealField(100)); Exp[Sqrt[2]/3]; // G. C. Greubel, Aug 19 2018

Maple

Digits := 85: evalf(exp(sqrt(2)/3)); # End program 1.
P := proc(n) : if n=0 then 1 else P(n) := expand((1/n)*(add(x(n-k)*P(k), k=0..n-1))) fi; end: x := proc(n): if n=3 then (sqrt(2)+1) else 1 fi: end: Digits := 56; evalf(P(120)); # End program 2.

Mathematica

First@ RealDigits@ N[Exp[Sqrt[2]/3], 85] (* Michael De Vlieger, Jun 27 2016 *)

PARI

my(x=exp(sqrt(2)/3)); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", ")) \\ Felix Fröhlich, Jun 27 2016

Formula

c = exp(sqrt(2)/3)

c = limit(P(n), n -> infinity) with P(n) = (1/n)*(sum(x(n-k)*P(k), k=0..n-1)) for n >= 1, and P(0) =1, with x(3) = (1 + sqrt(2)), the silver mean A014176, and x(n) = 1 for all other n.